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A203001 Symmetric matrix based on A007598, by antidiagonals.

%I A203001 #7 Jul 12 2012 00:39:53
%S A203001 1,1,1,4,2,4,9,5,5,9,25,13,18,13,25,64,34,41,41,34,64,169,89,113,99,
%T A203001 113,89,169,441,233,290,266,266,290,233,441,1156,610,765,689,724,689,
%U A203001 765,610,1156,3025,1597,1997,1811,1866,1866,1811,1997,1597,3025
%N A203001 Symmetric matrix based on A007598, by antidiagonals.
%C A203001 Let s=A007598 (squared Fibonacci numbers), and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s.  Let T' be the transpose of T.  Then A203001 represents the matrix product M=T'*T.  M is the self-fusion matrix of s, as defined at A193722.  See A203002 for characteristic polynomials of principal submatrices of M, with interlacing zeros.
%e A203001 Northwest corner:
%e A203001 1...1...4....9....25....64
%e A203001 1...2...5....13...34....89
%e A203001 4...5...18...41...113...290
%e A203001 9...13..41...99...266...724
%t A203001 s[k_] := Fibonacci[k]^2;
%t A203001 U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
%t A203001 L = Transpose[U]; M = L.U; TableForm[M]
%t A203001 m[i_, j_] := M[[i]][[j]];
%t A203001 Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
%t A203001 f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
%t A203001 Table[f[n], {n, 1, 12}]
%t A203001 Table[Sqrt[f[n]], {n, 1, 12}]   (* A001654 *)
%t A203001 Table[m[1, j], {j, 1, 12}]      (* A007598 *)
%t A203001 Table[m[2, j], {j, 1, 12}]      (* A001519 *)
%t A203001 Table[m[j, j], {j, 1, 12}]      (* A005969 *)
%Y A203001 Cf. A203002, A202453.
%K A203001 nonn,tabl
%O A203001 1,4
%A A203001 _Clark Kimberling_, Dec 27 2011